If $a\mid c$ and $b\mid c$, must $ab$ divide $c$?

Question

If $a\mid c$ and $b\mid c$, must $ab$ divide $c$? Justify your answer.

$a\mid c$, $c=ak$ for some integer $k$

$b\mid c$, $c=bu$ for some integer $u$

From here I wanted to try to check if there were counter examples I could use,

$c\ne(ab)w$ for some integer $w$

From here I got stuck because there is nothing I can plug into that equation so I know that I am probably missing something.

Answer 1

This is not true.

Take for instance $a=b=2$ and $c=2$.

Then $a\mid c $ and $b\mid c$, but

$$ab=4\nmid 2=c.$$

Answer 2

No it must not, you can give a simple counter-example like $3\mid 9$ and $9\mid 9$ but obviously $$ab=9\cdot3=27 \nmid9$$ Hope it helps you out!

Answer 3

$4\mid 12$ and $6\mid12$ but $4\times6\nmid12. \qquad$

The proposition is true when $\gcd(a,b)=1.$